It is not as easy of just switching parts. Even if the mobos are from the same OEM, different models sometimes have different drivers.
Now if you decide to migrate your mobo to your Dad's rig make sure that it is an upgrade and not a downgrade. Compare the two and see which is newer, has more features, etc.
If the processors are the same you may be able to migrate, but if I read your intent you plan to put a new mobo and processor in your own rig. If so I'd leave the processor in and migrate the mobo and processor together -- you know that they are compatible. If the processors are the same, I might even take Dad's CPU and put it on the new mobo for your rig. You don't have to take out the CPU, clean it, apply thermal grease, and put the cooler on twice that way.
In migrating your mobo you will probably have to reinstall your OS and all the native drivers for the other hardware. A new clean install is always best and usually improves performance by getting rid of all the old junk slowing your Dad's rig down.
You will, of course, have to reinstall your OS for your new mobo & CPU. You don't have to wipe your old HDD but you will have the clutter of the old OS install.